Multiply each term on both sides by \(co\sec{A}-1\).
\[\frac{1}{co\sec{A}-1}(co\sec{A}-1)-\frac{1}{co\sec{A}+1}(co\sec{A}-1)=2\tan^{2}A(co\sec{A}-1)\]
Cancel \(co\sec{A}-1\).
\[1-\frac{1}{co\sec{A}+1}(co\sec{A}-1)=2\tan^{2}A(co\sec{A}-1)\]
Simplify \(\frac{1}{co\sec{A}+1}(co\sec{A}-1)\) to \(\frac{co\sec{A}-1}{co\sec{A}+1}\).
\[1-\frac{co\sec{A}-1}{co\sec{A}+1}=2\tan^{2}A(co\sec{A}-1)\]
Expand.
\[1-\frac{co\sec{A}-1}{co\sec{A}+1}=2co\tan^{2}A\sec{A}-2\tan^{2}A\]
Subtract \(1\) from both sides.
\[-\frac{co\sec{A}-1}{co\sec{A}+1}=2co\tan^{2}A\sec{A}-2\tan^{2}A-1\]
Multiply both sides by \(co\sec{A}+1\).
\[-co\sec{A}+1=(2co\tan^{2}A\sec{A}-2\tan^{2}A-1)(co\sec{A}+1)\]
Expand.
\[-co\sec{A}+1=2{c}^{2}{o}^{2}\tan^{2}A\sec^{2}A+2co\tan^{2}A\sec{A}-2co\tan^{2}A\sec{A}-2\tan^{2}A-co\sec{A}-1\]
Simplify \(2{c}^{2}{o}^{2}\tan^{2}A\sec^{2}A+2co\tan^{2}A\sec{A}-2co\tan^{2}A\sec{A}-2\tan^{2}A-co\sec{A}-1\) to \(2{c}^{2}{o}^{2}\tan^{2}A\sec^{2}A-2\tan^{2}A-co\sec{A}-1\).
\[-co\sec{A}+1=2{c}^{2}{o}^{2}\tan^{2}A\sec^{2}A-2\tan^{2}A-co\sec{A}-1\]
Cancel \(-co\sec{A}\) on both sides.
\[1=2{c}^{2}{o}^{2}\tan^{2}A\sec^{2}A-2\tan^{2}A-1\]
Add \(2\tan^{2}A\) to both sides.
\[1+2\tan^{2}A=2{c}^{2}{o}^{2}\tan^{2}A\sec^{2}A-1\]
Add \(1\) to both sides.
\[1+2\tan^{2}A+1=2{c}^{2}{o}^{2}\tan^{2}A\sec^{2}A\]
Simplify \(1+2\tan^{2}A+1\) to \(2\tan^{2}A+2\).
\[2\tan^{2}A+2=2{c}^{2}{o}^{2}\tan^{2}A\sec^{2}A\]
Divide both sides by \(2\).
\[\frac{2\tan^{2}A+2}{2}={c}^{2}{o}^{2}\tan^{2}A\sec^{2}A\]
Factor out the common term \(2\).
\[\frac{2(\tan^{2}A+1)}{2}={c}^{2}{o}^{2}\tan^{2}A\sec^{2}A\]
Cancel \(2\).
\[\tan^{2}A+1={c}^{2}{o}^{2}\tan^{2}A\sec^{2}A\]
Divide both sides by \({o}^{2}\).
\[\frac{\tan^{2}A+1}{{o}^{2}}={c}^{2}\tan^{2}A\sec^{2}A\]
Divide both sides by \(\tan^{2}A\).
\[\frac{\frac{\tan^{2}A+1}{{o}^{2}}}{\tan^{2}A}={c}^{2}\sec^{2}A\]
Simplify \(\frac{\frac{\tan^{2}A+1}{{o}^{2}}}{\tan^{2}A}\) to \(\frac{\tan^{2}A+1}{{o}^{2}\tan^{2}A}\).
\[\frac{\tan^{2}A+1}{{o}^{2}\tan^{2}A}={c}^{2}\sec^{2}A\]
Divide both sides by \(\sec^{2}A\).
\[\frac{\frac{\tan^{2}A+1}{{o}^{2}\tan^{2}A}}{\sec^{2}A}={c}^{2}\]
Simplify \(\frac{\frac{\tan^{2}A+1}{{o}^{2}\tan^{2}A}}{\sec^{2}A}\) to \(\frac{\tan^{2}A+1}{{o}^{2}\tan^{2}A\sec^{2}A}\).
\[\frac{\tan^{2}A+1}{{o}^{2}\tan^{2}A\sec^{2}A}={c}^{2}\]
Use Trigonometric Identities.
\[\frac{\tan^{2}A+1}{{o}^{2}\tan^{2}A(\tan^{2}A+1)}={c}^{2}\]
Cancel \(\tan^{2}A+1\).
\[\frac{1}{{o}^{2}\tan^{2}A}={c}^{2}\]
Take the square root of both sides.
\[\pm \sqrt{\frac{1}{{o}^{2}\tan^{2}A}}=c\]
Switch sides.
\[c=\pm \sqrt{\frac{1}{{o}^{2}\tan^{2}A}}\]
c=sqrt(1/(o^2*tan(A)^2)),-sqrt(1/(o^2*tan(A)^2))
